{"id":"86b4945d-7eff-476a-b782-92fc7140f9dd","arxiv_id":"2607.08468","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For sufficiently large n, the maximum spectral radii of M_{k+1}-free and non-trivial intersecting 3-graphs on n vertices are determined and the extremal hypergraphs are characterized.","lead":"The paper determines the largest spectral radius of 3-uniform hypergraphs that either avoid a matching of size k+1 or are non-trivially intersecting, for all large enough vertex counts. It matters to extremal combinatorics because it lifts classical edge-count theorems (Erdős–Ko–Rado, Hilton–Milner, Erdős matching) into the spectral setting.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"Manuscript body absent; central spectral-extremal claims cannot be checked for gaps or correctness.","rationale":"The reader correctly observed that only the abstract and first paragraph are available, rendering the strongest claim unverifiable and forcing an UNVERDICTED verdict with low confidence. The same incompleteness is the single most load-bearing obstacle: without the body one cannot even begin to test the “sufficiently large n” hypothesis or the spectral comparisons that would support the claimed determination. No independent formal verification, code, or data exists to compensate. Once the full text is supplied the same stress-test can be re-run on the actual arguments; until then the reader’s assessment stands unchanged.","tokens_in":2695,"tokens_out":403,"duration_ms":14997,"concrete_test":"Retrieve the complete arXiv source or PDF of 2607.08468 and confirm that explicit theorems appear that (i) identify the extremal 3-graphs by name or construction and (ii) give a concrete (even if large) lower bound on n in terms of k; if either is missing or the proofs contain an unclosed asymptotic estimate, the determination claim remains unsubstantiated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The supplied text terminates after the opening sentences of §1; no theorem statements, no extremal constructions (e.g., the expected spectral analogues of the Erdős matching or Hilton–Milner hypergraphs), and no proofs appear. Consequently the assertion that the maximum spectral radii of M_{k+1}-free and of non-trivial intersecting 3-graphs are completely determined for all sufficiently large n rests on material that is simply not present. Any potential gap in the asymptotic analysis, any hidden dependence of the threshold n0(k) on unspecified constants, or any error in the comparison of spectral radii therefore cannot be located or ruled out.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper claims spectral analogues of the Erdős matching problem and the Hilton–Milner theorem for 3-uniform hypergraphs. Concretely, for all sufficiently large n it asserts that the maximum spectral radius of an M_{k+1}-free 3-graph on n vertices, and of a non-trivial intersecting 3-graph on n vertices, are completely determined, and that the extremal hypergraphs are characterized.","tokens_in":2834,"tokens_out":541,"duration_ms":19788,"significance":"If the claimed determinations and characterizations hold, the work would supply clean spectral counterparts to two classical extremal results in 3-uniform hypergraph theory, contributing to the growing literature that translates matching and intersecting-family theorems into spectral-radius statements. The asymptotic framing is standard for such problems; an explicit or reasonably effective threshold n0(k) would further increase the result’s utility.","major_comments":[{"comment":"The manuscript text supplied for review terminates after the opening sentences of §1 (Introduction). No theorem statements, extremal constructions, lemmas or proofs appear. The central claims—that the maximum spectral radii are completely determined and the extremal 3-graphs characterized—therefore cannot be checked for correctness, gaps in the asymptotic analysis, or hidden dependence of the threshold on unspecified constants.","section":null},{"comment":"Abstract and §1: the results are stated only for “sufficiently large n,” with no explicit or effective bound n0(k) provided in the available text. Without such a bound (or a clear indication of its order of magnitude), the claimed determination remains purely existential and its range of validity is unverifiable from the given material.","section":null}],"minor_comments":[{"comment":"Abstract: typographical errors “csae” (case), “hpergraph” (hypergraph).","section":null},{"comment":"AMS subject classifications line: “classiflcations” should be “classifications”.","section":null},{"comment":"Introduction opening: spacing and formatting of “Ann-vertexr-uniform” and similar compounds need correction for readability.","section":null}],"recommendation":"uncertain","confidential_remarks":"The arXiv extract and the cacheable source both end after the first paragraph of the Introduction; the submission package appears incomplete. The editor should request the full PDF (or confirm that the body was omitted in error) before any further refereeing cycle. Until the proofs are available the soundness of the spectral claims cannot be assessed."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know is that this paper claims the spectral-radius versions of the classical matching and non-trivial intersecting problems for 3-uniform hypergraphs: for large enough n it pins down the max spectral radius of M_{k+1}-free 3-graphs and of non-trivial intersecting 3-graphs, and names the extremal examples. That is real, incremental work inside the spectral extremal program; it is not a re-organization of the field, but specialists will want the statements on the shelf.\n\nWhat is new is the spectral determination itself. The edge-extremal results (EKR, Hilton–Milner, Erdős matching for r=3) are classical; the paper says it supplies the corresponding spectral maxima and characterizations. The abstract and opening are cleanly written and correctly locate the work relative to those theorems. No circularity is visible; the dependence on prior combinatorial results is ordinary.\n\nThe soft spot is structural and currently decisive: the manuscript extract we have stops after the first paragraph of the introduction. There are no theorem statements, no extremal constructions, and no proofs. So we cannot check the asymptotic analysis, the size of the “sufficiently large n” threshold, or the spectral-radius comparisons. That is not a manufactured flaw; it is simply that the load-bearing material is absent from what we were given. Once the full body is available those points can be re-scored; until then the claims remain unverified.\n\nThis is for people already working on spectral hypergraph extremal problems. A serious editor should send it to referees rather than desk-reject: the problem is standard, the claimed contribution is concrete, and the literature positioning is honest. I would not cite it yet because I cannot verify the proofs, but I would bring the abstract to a reading group as a pointer to a result we should watch for. Engage when the full text appears; do not treat the abstract as settled.","headline":"Spectral EKR/Hilton–Milner/matching for 3-graphs: solid specialist progress, but the full proofs are not in the extract we have, so the asymptotic claims stay unchecked.","tokens_in":3440,"tokens_out":500,"would_cite":false,"duration_ms":4995,"reading_group":"maybe","serious_thinker":"unclear","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["05C35","05C65"],"pacs":[],"model":"grok-4.5","headline":"For large n, the largest spectral radius among 3-uniform hypergraphs without a matching of size k+1, and among non-trivial intersecting ones, is achieved by two classical constructions.","keywords":["spectral radius","3-uniform hypergraph","matching","intersecting family","Hilton–Milner","Erdős matching conjecture","extremal hypergraph"],"falsifier":"Compute the spectral radius of the claimed extremal hypergraphs and of a carefully chosen competing family for moderate n (say n=50–100 and small k); if any competitor exceeds the claimed maximum, the asymptotic determination is false.","tokens_in":3568,"feed_emoji":"📐","tokens_out":576,"duration_ms":5481,"temperature":0.7,"pith_summary":"This paper translates two classical questions about the largest 3-uniform hypergraphs with restricted intersections into the language of spectral radius. One restriction forbids a matching of size k+1 (k pairwise disjoint edges); the other requires that every pair of edges meets, yet no single vertex lies in every edge. The authors prove that, once the number of vertices is large enough, the spectral radius is maximised precisely by the same two families that maximise the number of edges: the complete 3-graph on a set of size 3k-1 with an isolated vertex set, and the Hilton–Milner construction that forces every edge to meet a fixed pair. The extremal hypergraphs are completely characterised. A sympathetic reader cares because spectral radius often captures global expansion and connectivity more sharply than edge count alone; settling these spectral versions therefore gives a stronger extremal statement for the same combinatorial constraints.","feed_headline":"Spectral radius of matching-free 3-graphs is settled for large n","feed_subtitle":"Two classical constructions maximise the largest eigenvalue; both extremal hypergraphs are characterised.","key_machinery":"The spectral radius of a 3-uniform hypergraph (largest eigenvalue of its adjacency tensor), together with stability arguments that force any near-extremal hypergraph to be close in structure to one of the two classical constructions.","core_discovery":"For all sufficiently large n the maximum spectral radius of an n-vertex 3-uniform hypergraph with no matching of size k+1 is attained uniquely by the complete 3-partite Turán-type construction of order 3k-1 plus isolates, while the maximum spectral radius among non-trivial intersecting 3-graphs is attained uniquely by the Hilton–Milner hypergraph; both extremal objects are completely characterised.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Spectral max of M_{k+1}-free 3-graphs settled for large n","Max spectral radius for non-trivial intersecting 3-graphs fixed","Turán and Hilton-Milner 3-graphs maximise spectral radius","Extremal spectra of matching-free 3-uniform hypergraphs found","Spectral radius of intersecting 3-graphs characterised for large n"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The theorems are stated only for n larger than some unspecified threshold that depends on k; if that threshold is huge or the asymptotic analysis has a gap, the claimed determination fails for every practical size.","fun_headline_variants_meta":{"raw":{"variants":["Spectral max of M_{k+1}-free 3-graphs settled for large n","Max spectral radius for non-trivial intersecting 3-graphs fixed","Turán and Hilton-Milner 3-graphs maximise spectral radius","Extremal spectra of matching-free 3-uniform hypergraphs found","Spectral radius of intersecting 3-graphs characterised for large n"]},"model":"grok-4.5","effort":"low","cost_usd":0.004872,"raw_usage":{"total_tokens":1334,"prompt_tokens":735,"num_sources_used":0,"completion_tokens":81,"cost_in_usd_ticks":48720000,"prompt_tokens_details":{"text_tokens":735,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":518,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":735,"tokens_out":81,"duration_ms":5460,"temperature":1.0,"reasoning_tokens":518,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T06:58:38.624686+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute the spectral radius of the claimed extremal hypergraphs and of a carefully chosen competing family for moderate n (say n=50–100 and small k); if any competitor exceeds the claimed maximum, the asymptotic determination is false.","supporting_citations":[],"review_version":1}